A darktable style can take full advantage of every module in darktable, but the style can’t be used in other applications.
A LUT is color, it is literally a “look up table” or a mapping of one particular value to another particular value (more or less) LUTs can be used in any application that supports the particular LUT format. It useful doing color grading and other artistic type of things. You won’t get things like local contrast or a low pass filter with a LUT though (like you can with a style).